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Báo cáo hóa học: "UNIQUENESS RESULTS FOR ELLIPTIC PROBLEMS WITH SINGULAR DATA"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: UNIQUENESS RESULTS FOR ELLIPTIC PROBLEMS WITH SINGULAR DATA | UNIQUENESS RESULTS FOR ELLIPTIC PROBLEMS WITH SINGULAR DATA LOREDANA CASO Received 22 December 2005 Revised 22 May 2006 Accepted 12 June 2006 We obtain some uniqueness results for the Dirichlet problem for second-order elliptic equations in an unbounded open set o without the cone property and with data depending on appropriate weight functions. The leading coefficients of the elliptic operator are VMO functions. The hypotheses on the other coefficients involve the weight function. Copyright 2006 Loredana Caso. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let o be an open subset of R n 3. Consider in o the uniformly elliptic differential operator with measurable coefficients r n d2 d L - aij . a - a 1.1 i j 1 dxjdXj i 1 ỎXi and the Dirichlet problem Lu 0 u e W2 p O n W 1 p O D withp e 1 ro . Suppose that o verifies suitable regularity assumptions. If p n a j e L O i j 1 . n and the coefficients a i 1 . n a satisfy certain local summability conditions with a 0 then it is possible to obtain a uniqueness result for the problem D using a classical result of Alexandrov and Pucci see 17 for the case of bounded open sets and 6 Section 1 for the unbounded case . If p n some more assumptions on the a j s are necessary to get uniqueness results for the problem D . If o is bounded problem D has been widely studied by several authors under various hypotheses on the leading coefficients. In particular if the coefficients a j Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID 98923 Pages 1-9 DOI 10.1155 BVP 2006 98923 2 Uniqueness results for elliptic problems with singular data belong to the space Co U then uniqueness results for problem D have been obtained see 12-15 . On the other hand when the coefficients aij are required to be discontinuous the classical result by .